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\title{Mathematical Writing Exercise Chapter 3.1 - 3.7 }
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%\date{2023 年 10 月 31 日}
\date{April 2, 2024}

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\begin{enumerate}

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\item  %Problem 00
True or False. (3.0 English Usage - Introduction)
\begin{enumerate}[label={(\arabic*)}]
\item  
In this chapter I discuss {\color{red}aspects of English usage} that are particularly relevant to mathematical writing. 
%
You should keep three things in mind as you read this chapter. 
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\item  
First, on many matters of English usage {\color{red}rules have exceptions}, and, moreover, not all authorities agree on the rules. 
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I have consulted several usage guides (those described in 4.3) and have tried to give a view that reflects usage in writing today. 
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Second, about half the topics discussed here are not peculiar to the English language, but simply reflect {\color{red}common sense} in writing. 
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\item  
Third, many of the points mentioned are not vitally important when taken in isolation. 
%
But, as van Leunen explains (quoted in [205, p. 97]), 

\begin{center}\fbox{
\begin{minipage}{12cm}
Tone is important, and tone consists entirely of making these
tiny, tiny choices. 
If you make enough of them wrong ... then you won't get your maximum readership. 
The reader who has to read the stuff will go on reading it, but with less attention, less commitment than you want. 
And the reader who doesn't have to read will stop.
\end{minipage}
} \end{center}
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\end{enumerate}

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\item  %Problem 01
True or False. (3.1. A or An?)
\begin{enumerate}[label={(\arabic*)}]
\item  
Whether {\color{red}\it a} or {\color{red}\it an} should precede a noun depends on how the first syllable is pronounced: {\color{red}\it a} is used if the first syllable begins with a consonant sound and {\color{red}\it an} if it begins with a vowel sound. 
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\item  
For this rule, the initial ``yew'' sound in the words university and European is regarded as a consonant sound: thus ``a university'', `a European''. 
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\item  
For words beginning with h, {\color{red}\it a} is used unless the h is not sounded. 
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\item  
The only words in this last category are heir, honest, honor (UK honour), hour, and their derivatives. 
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\item  
The question ``{\color{red}\it a} or {\color{red}\it an}?'' most frequently arises with acronyms, initialisms, and proper nouns. 
%
An easy example is ``an NP-complete problem.'' 
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\item  
In the world of mathematical software, given that the usual pronunciation of LAPACK is l-a-pack and that of NAG is nag, we would write ``an LA-PACK routine'' but ``a NAG library routine''.
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\end{enumerate}

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\item  %Problem 02
True or False. (3.2. Abbreviations) 
\begin{enumerate}[label={(\arabic*)}]
\item  
This table shows {\color{red}some abbreviations of Latin words} commonly used in technical writing. 

\begin{center}
%\begin{table}
\begin{tabular}{lll} \hline 
Abbreviation & Latin expression & English meaning \\ \hline 
e.g. & exempli gratia & for example \\ \hline 
i.e. & id est & that is  \\ \hline 
etc. & et cetera & and so on \\ \hline 
cf. & confer & compare \\ \hline 
et al. & et alia, et alii, or et aliae & and others \\ \hline 
N.B., NB, n.b., or nb & nota bene & note \\ \hline 
\end{tabular}
%\end{table}
\end{center}

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\item  
One school of thought says that the use of the abbreviations {\color{red}\it e.g.} and {\color{red}\it i.e.} is bad style, and that {\color{red}\it for example} and {\color{red}\it that is} make for a smoother-flowing sentence. 
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\item  
In any case, all four forms should usually be followed by a comma and preceded by a comma, dash, semicolon, or 
parenthesis. 
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\item
When using the abbreviations you should type {\color{red}\it e.g.} and {\color{red}\it i.e.}, not {\color{red}\it eg.} or {\color{red}\it ie.}, since the abbreviations represent two words. 
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\item
The abbreviations {\color{red}\it i.e.} and {\color{red}\it etc.} are often used unnecessarily. 
%
For example: Delete ``i.e.,'' in `The most expensive method, i.e., Newton's method, converges quadratically.''
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\item
Another example: The sentence ``Fill-in can be avoided by using orderings such as reverse Cuthill-McKee, minimum degree, nested dissection, etc.'' should be ended with ``and nested dissection'', because ``such as'' is understood to introduce an incomplete list.
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\item
The less frequently used {\color{red}\it cf.} has only one period, because it is an abbreviation of a single word. 
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\item
Often, {\color{red}\it cf.} is used incorrectly in the sense of ``see'', as in ``cf. [6] for a discussion''.
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\item
In the abbreviation {\color{red}\it et al.} only the second word is abbreviated, so only one period is needed. ......
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\item
The abbreviation {\color{red}\it N.B.} is not often used in technical writing, probably because it has to appear at the beginning of a sentence and is somewhat inelegant. 
%
You can usually find a better way to give the desired emphasis. 
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\item
The abbreviation {\color{red}\it iff}, although handy in notes, is usually spelled out as {\color{red}\it if and only if}. 
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\item
An important abbreviation used in citations is {\color{red}\it p.} for page and {\color{red}\it pp.} for pages. 
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\item
The normal practice when introducing a nonstandard abbreviation or acronym is to spell out the word or phrase in full on its first occurrence and place the abbreviation immediately after it in parentheses. 
%
Thereafter the abbreviation is used. Example:
\begin{center}\fbox{
\begin{minipage}{12cm}
Gaussian elimination (GE) is a method for solving a system of n linear equations in n unknowns. GE has a long history; a variant of it for solving systems of three equations in three unknowns appears in the classic Chinese work ``Chiu-Chang Suan-Shu'', written around 250 B.C.
\end{minipage}
} \end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

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\item  %Problem 03
True or False. (3.3. Absolute Words) 
\begin{enumerate}[label={(\arabic*)}]
\item  
Certain adjectives have an absolute meaning and cannot be qualified by words such as {\color{red}\it less}, {\color{red}\it quite}, {\color{red}\it rather}, and {\color{red}\it very}. 
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\item
For example, it is wrong to write {\color{red}\it most unique} (replace by {\color{red}\it unique}, or perhaps {\color{red}\it most unusual}), {\color{red}\it absolutely essential}, {\color{red}\it more ideal}, or {\color{red}\it quite impossible}. 
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\item
However, {\color{red}\it essentially unique} is an acceptable term in mathematical writing: it means unique up to some known transformations. 
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\item
Some other words frequently used with an absolute sense are modified with a precise meaning in standard mathematical usage, such as ``convergence is {\color{red}\it almost certain}'' and ``the function is differentiate {\color{red}\it almost everywhere}''. 
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\end{enumerate}

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\item  %Problem 04
True or False. (3.4. Acronyms and Initialisms) 
\begin{enumerate}[label={(\arabic*)}]
\item  
Acronyms and initialisms are words formed from the initial letters, or groups of letters, of a series of words. 
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\item
The distinction is that {\color{red} acronyms are pronounceable words} whereas {\color{red} initialisms are pronounced letter by letter}. 
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\item
Examples of acronyms in mathematics and computing are JPEG (Joint Photographic Experts Group), MIMO (multi-input multi-output), NaN (not a number), GNU (GNU's not Unix - a recursive acronym), and SIAM (Society for Industrial and Applied Mathematics). 
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\item
Examples of initialisms are DCT (discrete cosine transform), PDE (partial differential equation), and SVD (singular value decomposition).
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\item
New acronyms and initialisms are regularly formed in research papers, often to denote methods or techniques. 
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\item
Nonnative speakers are advised to be careful in constructing them as the resulting words may have unsuspected meanings (two egregious examples can be found in [359]). 
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\item
These kinds of abbreviations should not be overused, otherwise the text becomes dominated by capital letters.
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\end{enumerate}

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\item  %Problem 05
True or False. (3.5. Active versus Passive) 
\begin{enumerate}[label={(\arabic*)}]
\item  
{\color{red} Prefer the active voice (``X did Y'') to the passive voice (``Y was done by X''). } 
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\item
The active voice adds life and movement to writing, whereas too much of the passive voice weakens the communication between writer and reader. 
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\item
\begin{enumerate}
\item
\underline{Passive}: The answer was provided to sixteen decimal places by Gaussian elimination. 
 \item
\underline{Active}: Gaussian elimination gave the answer to sixteen decimal places.
\end{enumerate}
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\item
\begin{enumerate}
\item
\underline{Passive}: The failure of Newton's method to converge is attributed to the fact that the Jacobian is singular at the
solution.
\item 
\underline{Active}: Newton's method fails to converge because the Jacobian is singular at the solution. 
\end{enumerate}
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\item
\begin{enumerate}
\item
\underline{Passive}: A numerical example is now given to illustrate theabove result. 
\item
\underline{Active}: We give a numerical example to illustrate this result. 
\item
\underline{Active}: The following numerical example illustrates this result.
\end{enumerate}
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\item
The second example in the following {\color{red}trio} illustrates a further degree of abstraction in which a verb is replaced by an bstract noun modified by another verb. 

\begin{enumerate}
\item
\underline{Passive}: The optimality of $y$ was verified by checking that the Hessian matrix was positive definite. 

\item 
\underline{Passive and indirect}: Verification of the optimality of $y$ was achieved by checking that the Hessian matrix was positive definite. 

\item 
\underline{Active}: We verified the optimality of $y$ by checking that the Hessian matrix was positive definite.
\end{enumerate}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item
Other ``was'' phrases that can often be removed by rewriting are ``was performed'', ``was experienced'', ``was carried out'', ``was conducted'', and ``was accomplished''. 
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\item
{\color{red} The passive voice does have an important role to play, however. } 
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\item
It adds variety, may be needed to put emphasis on a certain part of a sentence, and may be the only choice if specific information required for an active variant is unknown or inappropriate to mention. 
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\item
Examples where the passive voice {\color{red} allows the desired emphasis} are ``An ingenious proof of this conjecture was constructed by C. L. Ever,'' and (from the writings of Paul Halmos [288, p. 96]) ``The subjects just given honorable mention, as well as the three actually discussed in detail, have been receiving serious research attention in the course of the last twenty years.'' ......
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\item 
The passive voice is also {\color{red} useful for euphemistic effect}, allowing the clumsy experimenter to say ``The 
specimen was accidentally strained during mounting'' instead of ``I dropped the specimen on the floor.''
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

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\item  %Problem 06
True or False. (3.6. Adjective and Adverb Abuse) 
\begin{enumerate}[label={(\arabic*)}]
\item  
It is easy to overuse adverbs and adjectives. 
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\item 
They should be used sparingly, to add precision to a sentence.
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\item 
Use an adverb to change the meaning of a verb, not to reinforce a meaning that it already has. 
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\item 
For example, in ``The uniqueness of these solutions is {\color{red}completely} open due to the lack of $L^\infty$ {\color{red} a priori} estimates'' the adverb ``{\color{red}completely}'' should be omitted, as there are not degrees of openness.
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\item 
In ``The proof is very easy'' and ``This inequality is quite important'' the adverbs are, again, best omitted. 
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\item 
Examples of acceptable usage are ``These results are very similar to those of Smith'' and ``This bound can be very
weak.'' 
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\item 
Sometimes you can rewrite the sentence {\color{red}to make the adverbial phrase more precise}. 
\begin{enumerate}
\item 
\underline{Original}: The numerical experiments show that the algorithm produces very accurate results. ......
\item 
\underline{Better}: The numerical experiments show that the algorithm produces results with errors at the level of the unit roundoff. 
\end{enumerate}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item 
An adverb that is especially overworked in mathematical writing is {\color{red}\it essentially}. 
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\item 
A valid use is in the expression ``essentially the same as'', which by convention in scientific writing means ``the same, except for minor details''.
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\item 
But it is often used with a vague sense that would be better made more precise.
\begin{enumerate}
\item 
\underline{Bad}: Beltrami (1873) essentially derived the singular value decomposition. 
\item 
\underline{Good}: Beltrami (1873) derived the singular value decomposition for square, nonsingular matrices. 
\end{enumerate}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item 
Sir Ernest Gowers notes ``It has been wisely said that the adjective is the enemy of the noun'' [122, p. 88]. 
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\item 
He suggests using adjectives to denote kind rather than degree. 
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\item 
In ``the {\color{red}nice} proof of the theorem'', ``{\color{red}nice}'' can be replaced by ``{\color{red}short}'', ``{\color{red}ingenious}'', ``{\color{red}elementary}'', or some other adjective that indicates what is {\color{red}nice} about the proof. 
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\item 
In the sentence ``It is {\color{red}interesting} to re-prove this result using Laplace transforms,'' {\color{red}\it instructive} is probably the intended word.
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\end{enumerate}

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\item  %Problem 07
True or False. (3.7. -age, -al, and -ology) 
\begin{enumerate}[label={(\arabic*)}]
\item  
Certain words that can be extended with an -age, -al, or -ology ending are often misused in the extended form. 
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\item 
The suffix tends to give a more abstract meaning, which makes it more difficult to use the word correctly. 
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\item 
For  example, {\color{red}\it usage} means a manner of using, so correct {\color{red}usage} is illustrated by ``in the original {\color{red}usage} the conjugate gradient method was not preconditioned'' and ``the {\color{red}use} of Euler's method is not recommended for stiff differential
equations.''
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\item 
An example where an -al ending is used incorrectly is ``the most {\color{red}pragmatical} opinion is the one expressed by the term's inventor'', in which the third word should be ``{\color{red}pragmatic}''.
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\item 
In the sentence ``A {\color{red}methodology} for determining the stability of the differential equation'' it is probably ``{\color{red}method}'' that is meant, as {\color{red}methodology} refers to the principles for deriving {\color{red}methods}.
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\end{enumerate}

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\end{enumerate}


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